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I've based the title on this question here. There, for sheaves $\mathcal{F}$ and $\mathcal{G}$ on toplogical spaces $X$ and $Y$ respectively, given a continuous map $\varphi:X\to Y$ they construct the unit and counit of the adjunction between $\varphi_*$ and $\varphi^{-1}$ via the universal properties of the various structures involved.

To try to get a feel for all the different constructions, I've been trying to write down the unit and counit explicitly by defining the actual maps themselves. This has been my attempt so far:

First we want to define $\varepsilon:\varphi^{-1}\varphi_*\mathcal{F}\to\mathcal{F}$. Let $U\subseteq X$ be open. For a sheaf $\mathcal{H}$ on $Y$, we define $\varphi^{-1}\mathcal{H}$ by sheafifying $\varphi_0^{-1}\mathcal{H}$, the presheaf given by setting $\varphi_0^{-1}\mathcal{H}$ to be the direct limit of $\mathcal{H}(V)$ for open $V\subseteq Y$ with $\varphi(U)\subseteq V$.

Elements of $\varphi_0^{-1}\varphi_*\mathcal{F}(U)$ are of the form $(a,V)$, where $V\subseteq Y$ is open, $U\subseteq\varphi^{-1}(V)$, and $a\in\mathcal{F}(\varphi^{-1}(V))$.

Let $\mathcal{H}=\varphi^{-1}_0\varphi_*\mathcal{F}$. Elements of $\varphi^{-1}\varphi_*\mathcal{F}(U)$ are then of the form $s:U\to\coprod_{u\in U}\mathcal{H}_u$, such that for any $u\in U$ we have $s(u)\in\mathcal{H}_u$ and an open set $W_u\subseteq U$ with $u\in W_u$ and some $a_u\in\mathcal{H}(W_u)$ such that, for any $w\in W_u$, the germ $(a_u)_w\in\mathcal{H}_w$ (this is just the definition of sheafification).

Now, $(a_u)_w$ is represented by $(b_{u,w},W'_{u,w})$ for some $W'_{u,w}\subseteq W_u$ with $w\in W'_{u,w}$ and $b_{u,w}\in\mathcal{H}(W'_{u,w})$. Then $b_{u,w}$ is of the form $(c_{u,w},V_{u,w})$, where $V_{u,w}\subseteq Y$ is open, $W'_{u,w}\subseteq\varphi^{-1}(V_{u,w})$, and $c_{u,w}\in\mathcal{F}(\varphi^{-1}(V_{u,w}))$.

Then define $\varepsilon_U:\varphi^{-1}\varphi_*\mathcal{F}(U)\to\mathcal{F}(U)$ by letting $\varepsilon_U(s)$ be given by gluing the $c_{u,u}|_{W'_{u,u}}$ together.


We now want to define $\eta:\mathcal{G}\to\varphi_*\varphi^{-1}\mathcal{G}$. Let $V\subseteq Y$ be open, and let $\mathcal{H}=\varphi_0^{-1}\mathcal{G}$. Then elements of $\varphi_*\varphi^{-1}\mathcal{G}(V)$ are of the form $s:\varphi^{-1}(V)\to\coprod_{u\in\varphi^{-1}(V)}\mathcal{H}_{\varphi^{-1}(V)}$, such that for any $u\in\varphi^{-1}(V)$ we have $s(u)\in\mathcal{H}_u$ and an open set $W_u\subseteq V$ with $u\in\varphi^{-1}(W_u)$ and some $a_u\in\mathcal{H}(\varphi^{-1}(W_u))$ such that, for any $w\in\varphi^{-1}(W_u)$, the germ $(a_u)_w\in\mathcal{H}_w$.

Now, $(a_u)_w$ is represented by $(b_{u,w},\varphi^{-1}(W'_{u,w}))$ for some $W'_{u,w}\subseteq W_u$ with $w\in\varphi^{-1}(W'_{u,w})$ and $b_{u,w}\in\mathcal{H}(\varphi^{-1}(W'_{u,w}))$. Then $b_{u,w}$ is of the form $(c_{u,w},V_{u,w})$, where $V_{u,w}\subseteq Y$ is open, $\varphi(\varphi^{-1}(W'_{u,w}))\subseteq V_{u,w}$, and $c_{u,w}\in\mathcal{G}(V_{u,w})$.

Let $a\in\mathcal{G}(V). $Then define $\eta_V:\mathcal{G}(V)\to\varphi_*\varphi^{-1}\mathcal{G}(V)$ by letting $\eta_V:a\mapsto(s:u\mapsto (a)_u)$, or to be precise unraveling the definitions (I think) and picking an explicit representative in the various equivalence classes $$s:u\mapsto ((a,V),\varphi^{-1}(V))$$

I feel like these definitions should be correct, since I don't think I could have made any other choices. However I'd like to be certain before I check they satisfy the naturality requirements etc.

Then really I have two questions:

  1. Are these definitions correct?
  2. Do people often use these definitions directly? There are so many equivalence classes of equivalence classes it can be difficult to picture/remember what is going on, especially when looking at stalks of inverse images. I know there's a more down to earth description of sheafification for sheaves of functions, but it still seems unwieldy. Then do we just tend to use the universal properties from the adjunction in practice?

Any help would be much appreciated.

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