The power and the copower of an object by a set

Let $\mathcal{M}$ be a locally small category (Small, locally small, and large categories, Category, object, morphism, domain, codomain, identity, composition, and hom-collection), let $c$ be an object of $\mathcal{M}$ and let $S$ be a set. Write $\mathbf{1}$ for the category with one object and only its identity morphism, and $D_c:\mathbf{1}\to \mathcal{M}$ for the diagram picking out $c$. A natural transformation between two functors $\mathbf{1}\to \mathcal{A}$ is a single morphism in the target category $\mathcal{A}$ between their values, since the only naturality equation is at an identity. When $\mathcal{A}=\mathbf{S}\mathbf{e}\mathbf{t}$ that morphism is a function.

The power of $c$ by $S$, written $c^S$ or $S⋔c$, is the weighted limit of the one-object diagram at the constant weight $S$ (Set-weighted limits and colimits): an object with a bijection

$\mathcal{M}(m,c^S)\cong \mathbf{S}\mathbf{e}\mathbf{t}(S,\mathcal{M}(m,c))$

natural in $m$, where the right-hand side is the set of functions $S\to \mathcal{M}(m,c)$ (The set $B^A$ of all functions $A\to B$, Sets and functions form the large locally small category $\mathbf{S}\mathbf{e}\mathbf{t}$, The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category).

The copower of $c$ by $S$, written $S\cdot c$, is the corresponding weighted colimit: an object with a bijection

$\mathcal{M}(S\cdot c,m)\cong \mathbf{S}\mathbf{e}\mathbf{t}(S,\mathcal{M}(c,m))$

natural in $m$. Neither object need exist.

The counit of the power is the family ${\mathrm{p}\mathrm{r}}_s:c^S\to c$ indexed by $s\in S$, obtained by applying the bijection to the identity of $c^S$; dually the copower carries injections ${\mathrm{i}\mathrm{n}}_s:c\to S\cdot c$.

Both are instances of the weighted limit and colimit of a diagram on a one-object index category, so nothing new is being defined: what is new is only the name and the notation, and the reason for having them is that the two constructions occur constantly once weights are allowed.

The enriched literature calls these the cotensor and the tensor of an object by an object of the base. Those names belong with the enriched development and are not used here; the $\mathbf{S}\mathbf{e}\mathbf{t}$-enriched names power and copower are the ones in force on this page.