This is an introductory post about some easy examples of internal Hom.
Definition. Let 
be a symmetric monoidal category, i.e. a category 
with a functor 
that is associative, unital, and commutative up to natural isomorphism. Then an internal Hom in is a functor
![\[\mathbf{Hom}(-,-) \colon \mathscr C\op \times \mathscr C \to \mathscr C\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-c14c9c14c53444e8e11893f5016cfc5a_l3.svg "Rendered by QuickLaTeX.com")
such that 
is a left adjoint to 
for any 
, i.e. there are functorial isomorphisms
![\[\operatorname{Hom}(X \otimes Y, Z) \stackrel\sim\to \operatorname{Hom}(X, \mathbf{Hom}(Y,Z)).\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-a9e9f03ab6dd48790300b0bb2a587c9a_l3.svg "Rendered by QuickLaTeX.com")
Remark. In the easiest examples, we typically think of 
as ‘upgrading 
to an object of ‘:
Example. Let 
be a commutative ring, and let 
be the category of -modules, with 
the tensor product. Then 
with its natural -module structure is an internal Hom, by the usual tensor-Hom adjunction:
![\[\operatorname{Hom}_R(M \otimes_R N, K) \cong \operatorname{Hom}_R(M, \mathbf{Hom}(N, K)).\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-c053d9b00b75a3814c80d88d9e6e646b_l3.svg "Rendered by QuickLaTeX.com")
The same is true when 
is the category of 
-bimodules for a not necessarily commutative ring .
However, we cannot do this for left -modules over a noncommutative ring, because there is no natural -module structure on 
for left -modules 
and 
. In general, the tensor product takes an 
-bimodule and a 
-bimodule and produces an 
-bimodule 
. Taking 
gives a way to tensor a right -module with a left -module, but there is no standard way to tensor two left -modules, let alone equip it with the structure of a left -module.
Example. Let 
. Then 
is naturally a set, making it into an internal Hom for 
:
![\[\operatorname{Hom}(X \times Y, Z) \stackrel\sim\to \operatorname{Hom}(X, \mathbf{Hom}(Y,Z)).\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-724d0507fa56e05fa53fbcc4a2af891d_l3.svg "Rendered by QuickLaTeX.com")
When is the categorical product 
, the internal 
(if it exists) is usually called an exponential object, in analogy with the case above.
Example. Another example of exponential objects is from topology. Let 
be the category of locally compact Hausdorff topological spaces. Then the compact-open topology makes 
into an internal Hom of topological spaces. (There are mild generalisations of this beyond the compact Hausdorff case, but for an arbitrary topological space 
the functor 
does not preserve colimits and hence cannot admit a right adjoint.)
Example. An example of a slightly different nature is chain complexes: let be a commutative ring, and let 
be the category of cochain complexes
![\[\ldots \to C^{i-1} \to C^i \to C^{i+1} \to \ldots\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-b11be861c86069e855982141f08aae85_l3.svg "Rendered by QuickLaTeX.com")
of -modules (meaning each 
is an -module, and the 
are -linear maps satisfying 
). Homomorphisms 
are commutative diagrams
![\[\begin{array}{ccccccc}\ldots & \to & C^i & \to & C^{i+1} & \to & \ldots \\ & & \!\!\!\!\! f^i\downarrow & & \downarrow f^{i+1}\!\!\!\!\!\!\! & & \\ \ldots & \to & D^i & \to & D^{i+1} & \to & \ldots,\!\!\end{array}\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-82b0b03f9e235983432f4ceb02a11492_l3.svg "Rendered by QuickLaTeX.com")
and the tensor product is given by the direct sum totalisation of the double complex of componentwise tensor products.
There isn’t a natural way to ‘endow 
with the structure of a chain complex’, but there is an internal Hom given by
![\[\mathbf{Hom}(C, D)^i = \prod_{m \in \mathbf Z} \operatorname{Hom}(C_m, D_{m+i}),\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-3cf574776217419a8383f65181d32a09_l3.svg "Rendered by QuickLaTeX.com")
with differentials given by
![\[d^if = d_D f - (-1)^i f d_C.\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-4c553e282de32891abf5bf1c6139c14c_l3.svg "Rendered by QuickLaTeX.com")
Then we get for example
![\[\operatorname{Hom}(R[0], \mathbf{Hom}(C, D)) \cong \operatorname{Hom}(C, D),\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-ced2185b01ff3da3f98eb6a02ef28dc1_l3.svg "Rendered by QuickLaTeX.com")
since a morphism ![R[0] \to \mathbf{Hom}(C, D)](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-42d0e04f02f937168abd66640fe06dce_l3.svg "Rendered by QuickLaTeX.com")
is given by an element 
such that 
, i.e. 
, meaning that 
is a morphism of cochain complexes.
Example. The final example for today is presheaves and sheaves. If is a topological space, then the category 
of abelian sheaves on has an internal Hom given by
![\[\mathbf{Hom}(\mathscr F, \mathscr G)(U) = \operatorname{Hom}(\mathscr F|_U, \mathscr G|_U),\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-9a82660bf3604fb7765d2d8b4e78a5cc_l3.svg "Rendered by QuickLaTeX.com")
with the obvious transition maps for inclusions 
of open sets. This is usually called the sheaf Hom. A similar statement holds for presheaves.
These are wonderful examples! Another of my favorites is the internal hom in the preorder of propositional logic, where the objects are propositions and a morphism from P to Q exists if and only if you can deduce Q from P. Then the internal hom-object is the proposition “P implies Q”! (The symmetric monoidal structure is “and”.) There’s a lovely parallel between the hom-object for propositions being “(not P) or Q” and the hom-object for vector spaces being “(V dual) tensor W”, as if in either case to turn A into B you just get rid of A and then tack on B.
P.S. A nice application of the Yoneda lemma is to prove that the isomorphisms of sets in your second centered equation can be promoted to internal isomorphisms between hom-objects! Something I’ve always wondered about is the relationship between a symmetric monoidal category having internal homs and being enriched over itself. Do you know what the story is there?
Nice example! As for your question, according to this MO question the answer is yet: symmetric closed monoidal categories (i.e. symmetric monoidal categories that have an internal hom) are always enriched over themselves. (There seems to be a statement in the non-symmetric case, but this does not apply to any of the examples above.)
Thanks! What the MO question asks but doesn’t get answered is to what extent the correspondence is one-to-one: if a symmetric monoidal category is enriched over itself, does it also gain an internal hom? It seems to get some similar data, but the last time I tried to write down all the axioms it didn’t seem quite to match up.
Okay, one more example of internal homs: any commutative monoid M can be promoted to a symmetric monoidal discrete category, and then one can ask whether that category has internal homs. The answer: yes, iff M is a group! If I remember correctly, that even works if M is not commutative, and one uses the suitably non-symmetric definition of monoidal closed category.