As I'm learning the basics of category theory I've noticed the idea of "fixing" or "picking" an object comes up often when defining some construction. For example:
for any bifunctor $F : C \times D \rightarrow E$ if we "fix" F with an object $c \in C$ then we have a functor $F c : D \rightarrow E$
if we "fix" an object $c \in C$ then functors $\times c : C \rightarrow C$ and $(-)^c : C \rightarrow C$ are adjoint: $\times c \dashv (-)^c$
It seems to me that in both the above examples whenever we "fix" an object we could select any object therefore if we consider all possible selections of $c \in C$ then would (1) could be extended to a "collection" of $F$s (one for each object in $C$ and (2) would extend to some "collection" of adjunctions (again one for each object in $C$)?
Do we have a way of encoding these "collections" in category theory?
More generally do we have a more technical way of describing things in category theory that vary over some fixed object?
My first intuition is to use functors from the single object category $1$ as a way of encoding the idea of "picking" a single object...but I'm not sure how to extend this idea to things like "picking an adjunction" (maybe some kind of category of adjunctions?)
For the first question :
Note that any functor(bifunctor) $F : C \times D \rightarrow E$ is precisely the same the same as a functor $F_{*} : C \rightarrow [D, E]$ sending object $c \in C$ to the object (functor) $F(c,-) : D \rightarrow E$ and a morphism $c \xrightarrow{f} c'$ onto the arrow (natural transformation) $F(f,1\_) : F(c,-) \Rightarrow F(c',-)$.
Likewise, you can consider $F$ as a functor $F^{*} : D \rightarrow [C,E]$.
What you really have here is a pair of isomorphisms of categories $$[C \times D, E] \cong [C, [D,E]] \cong [D, [C,E]]$$
For the second question:
You can interpret this "collection of adjunctions" in terms of the parametrized adjunction theorem. Assuming that $C$ has finite products, note that the product ${\_}\times {\_} : C \times C \rightarrow C$ is a functor(bifunctor). Further assuming that $C$ is Cartesian closed (that is each ${\_} \times c : C \rightarrow C$ has a right adjoint $(-)^c$), the collection of right adjoints themselves assemble to give a functor (bifunctor) $(-)^{-} : C^{op} \times C \rightarrow C$, which is more commonly known as the internal Hom of the category $C$.
One of the uses of the internal Hom functor is that it allows you to enrich your category $C$ over itself. Namely, $(C, \times, 1)$ already has the structure of a Cartesian monoidal closed category. You can now consider $C$ as a $(C, \times, 1)$-category $\mathbf{C}$ with $Ob(\mathbf{C}) = Ob(C)$ and $\mathbf{C}(x,y) = y^{x}$. The composition $z^y \times y^x \rightarrow z^x$ is the conjugate under the adjunction bijection, of the composition $$z^y \times y^x \times x \xrightarrow{1_{z^y} \times e} z^y \times y \xrightarrow{e} z$$ where $e$ is the counit of the adjunction ${\_} \times c \dashv (-)^c$ (denoted by the same letter $e$ for every object $c$). The identity arrow $1 \rightarrow c^c$ on an object $c \in \mathbf{C}$ is the conjugate under the adjunction bijection, of the identity arrow $$1 \times c \cong c \xrightarrow{1_c} c.$$