Let $I$ and $K$ be two categories. Let us consider two functors from $I$ to $K$:
- a covariant functor ${\mathcal F}:I\to K$, and
- a contravariant functor ${\mathcal G}:I\to K$,
and suppose that for each morphism $\varphi:A\to B$ in $I$ the following equalities hold: $$ {\mathcal F}(A)={\mathcal G}(A),\quad {\mathcal F}(B)={\mathcal G}(B), $$ $$ {\mathcal G}(\varphi)\circ {\mathcal F}(\varphi)=1_{{\mathcal F}(A)} $$ In other words, ${\mathcal G}(\varphi)$ is a retraction for ${\mathcal F}(\varphi)$ (and ${\mathcal F}(\varphi)$ is a coretraction for ${\mathcal G}(\varphi)$).
Question:
what is such a pair of functors $({\mathcal F},{\mathcal G})$ called?
Let me give an example that I have in mind. Suppose $G$ is an LP-group, i.e. a locally compact group which is a projective limit of its Lie quotient groups. Denote by $\lambda(G)$ the system of compact normal subgroups $H$ in $G$ such that the quotient group $G/H$ is a Lie group. Then, $$ G=\projlim_{H\in\lambda(G)}G/H. $$ We endow $\lambda(G)$ with the partial order $$ H\le L\quad\Leftrightarrow\quad H\supseteq L, $$ and this allows to consider $\lambda(G)$ as a category.
Now let us consider the space ${\mathcal C}(G)$ of continuous functions on $G$ and for each group $H\in\lambda(H)$ let us denote by ${\mathcal C}(G:H)$ the subspace of functions in ${\mathcal C}(G)$ which are invariant with respect to the shifts on elements of $H$:
$$
u(g\cdot h)=u(g)=u(h\cdot g),\qquad g\in G, \ h\in H
$$
We can note that
$$
{\mathcal C}(G:H)\cong{\mathcal C}(G/H),\qquad H\in\lambda(G).
$$
Now we can define the functor ${\mathcal F}$ as the one that turns $H\in\lambda(H)$ into the space ${\mathcal C}(G:H)$ and the pair $H\le L$ into the natural emdedding
$$
{\mathcal C}(G:H)\subseteq {\mathcal C}(G:L)
$$
And the functor ${\mathcal G}$ again turns $H\in\lambda(H)$ into the space ${\mathcal C}(G:H)$, but the pair $H\le L$ is turned into the natural projection
$$
{\mathcal C}(G:H)\gets {\mathcal C}(G:L)
$$
that turns each function $u\in{\mathcal C}(G:L)$ into its mean value with respect to the subgroup $H$:
$$
{\mathcal G}(H,L)(u)(g)=\int_H u(g\cdot h) \ \mu_H(d h),\qquad g\in G
$$
(here $\mu_H$ is the normalized Haar measure on $H$).
I am interested in this construction because it is used in Harmonic Analysis for proving some untrivial results, for example François Bruhat in his paper of 1961 uses this to prove that the space ${\mathcal D}(G)$ of test functions on a locally compact group $G$ is complete. (But he replaces everywhere ${\mathcal C}(G)$ by ${\mathcal D}(G)$.)
I thought that perhaps this construction has a special name in category theory since it looks very natural.
Remark. I used the term "coretraction-retraction" in the title, but I think it's worth saying that this is not a "true coretraction-retraction pair", since this is not a situation when the composition of functors gives the identity functor: we can't write $$ {\mathcal G}\circ {\mathcal F}=\operatorname{id}_I. $$
P.S. I also asked this at MathOverFlow.
Why isn't it just one functor?
Consider the category $\mathsf{Top}_{mr}$ consisting of topological spaces and morphisms $f:X \rightarrow Y$ given by embedding retraction pairs $X\overset{m}{\rightarrow} Y \overset{r}{\rightarrow} X$ (note both $m$ and $r$ are part of the datum of $f$!).
You define a functor $\mathcal{C}(G:-):\lambda(G) \rightarrow \mathsf{Top}_{mr}$, which sends $H\in\lambda(G)$ to $\mathcal{C}(G:H)=\mathcal{C}(G/H)$ and which sends any inclusion $L \subseteq H$ to the embedding-retraction pair $\mathcal{C}(G:H) \subseteq \mathcal{C}(G:L) \overset{avg}{\rightarrow} \mathcal{C}(G:H)$.
Note that you can still obtain your functors $\cal F$ and $\cal G$ by composing $\mathcal{C}(G:-)$ with the projections $\mathsf{Top}_{mr} \rightarrow \mathsf{Top}$ (which forgets the retractions) and $\mathsf{Top}_{mr} \rightarrow \mathsf{Top}^\text{op}$ (which forgets the embeddings).