\[\newcommand{\rv}[1]{\mathbf{#1}} \newcommand{\x}{\rv x} \newcommand{\y}{\rv y} \newcommand{\bar}[1]{\overline{#1}} \newcommand{\wtil}[1]{\widetilde{#1}} \newcommand{\what}[1]{\widehat{#1}} \newcommand{\ep}{\varepsilon} \newcommand{\ph}{\varphi} \newcommand{\maps}{\colon} \newcommand{\to}{\rightarrow} \newcommand{\xraw}{\xrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\To}{\Rightarrow} \renewcommand{\dot}{\centerdot} \renewcommand{\tensor}{\otimes} \newcommand{\pr}{^\prime} \newcommand{\op}[1]{#1^{\mathrm{op}}} \newcommand{\hom}{\textit{Hom}} \newcommand{\oo}{\circ} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\str}[1]{#1^{*}} \newcommand{\intvl}{\mathbb{I}} \newcommand{\yon}{\mathcal{Y}} \newcommand{\topos}{\mathfrak{Top}} \newcommand{\bTopos}{\mathfrak{BTop}} \newcommand{\BTop}{\mathfrak{BTop}} \newcommand{\con}{\mathfrak{Con}} \newcommand{\pos}{\mathfrak{Poset}} \newcommand{\topl}{\mathbb{Top}} \newcommand{\grpd}{\mathbb{Grpd}} \newcommand{\ab}{\mathbf{Ab}} \newcommand{\thT}{\mathbb{T}} \newcommand{\mod}{\mathbf{Mod}} \newcommand{\th}{\mathbf{Th}} \newcommand{\cl}{\mathbf{Cl}} \newcommand{\ob}{\mathbf{Ob}} \newcommand{\aut}{\mathbf{Aut}} \newcommand{\bun}{\mathbf{Bun}} \newcommand{\geom}{\mathbf{Geom}} \def\catg{\mathop{\mathcal{C}\! {\it at}}\nolimits} \def\Cat{\mathop{\mathfrak{Cat}}} \def\Con{\mathop{\mathfrak{Con}}} \def\CAT{\mathop{\mathbf{2} \mathfrak{Cat}}} \def\Cat{\mathop{\mathfrak{Cat}}} \newcommand{\Topos}{\mathfrak{Top}} \newcommand{\ETopos}{\mathcal{E}\mathfrak{Top}} \newcommand{\BTopos}{\mathcal{B}\mathfrak{Top}} \newcommand{\GTopos}{\mathcal{G}\mathfrak{Top}} \newcommand{\Psh}{\textit{Psh}} \newcommand{\Sh}{\textit{Sh}} \newcommand{\psh}[1]{\textit{Psh}(\cat{#1})} \newcommand{\sh}[1]{\textit{Sh}(\cat{#1})} \newcommand{\Id}{\operatorname{Id}} \newcommand{\Ho}{\operatorname{Ho}} \newcommand{\ad}{\operatorname{ad}} \newcommand{\Adj}{\operatorname{Adj}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Set}{\operatorname{Set}} \newcommand{\Pull}{\operatorname{Pull}} \newcommand{\Push}{\operatorname{Push}} \newcommand{\dom}{\operatorname{dom}} \newcommand{\cod}{\operatorname{cod}} \newcommand{\fun}{\operatorname{Fun}} \newcommand{\colim}{\operatorname{Colim}} \newcommand{\cat}[1]{\mathcal{#1}} \newcommand{\scr}[1]{\mathscr{#1}} \newcommand{\frk}[1]{\mathfrak{#1}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\CS}{\mathcal{S}} \newcommand{\CE}{\mathcal{E}} \newcommand{\CF}{\mathcal{F}}\]Warning: Incomplete
Toposes, geometric morphisms, and natural transforamtions form a 2-category denoted by $\mathfrak{Top}$. Note that for each pair of toposes $D$ and $E$, $\mathbf{Geom}(D,E)$ is a large though locally small category.
Example.
Suppose $X$ is a (non-$T_1$) topological space We define the following (non-trivial) partial order on points of $X$. \begin{equation} x \leq x’ \ \ \text{iff every neighbourhood of} \ \ x \ \ \text{contains} \ \ x’ \end{equation}
We can extend this order to all maps between topological spaces. Suppose $f,g: X \rightrightarrows Y$ are (continuous) maps. Define
\begin{equation}\label{def-order on maps} f \leq g \ \ \text{iff} \ \ f(x) \leq g(x) \ \ \text{for every} \ \ x \in X \end{equation}
A ramification of above defintion, which is straightforward to see, is that $f \leq g$ if and only if $f^{-1} (V) \subseteq g^{-1}(V)$ for evey open set $V$ of $Y$. Notice that $f$ and $g$ give us $f_{\ast},g_{\ast}: Sh(X) \rightrightarrows Sh(Y)$ which are pushfoward geometric morphisms along $f$ and $g$ respectively.
Thus, if $f \leq g$, then for any sheaf $P$ on $X$, and any open $V$ of $Y$, we get the restriction \(g_{\ast}(P)(V)=P(g^{-1}(V)) \rightarrow P(f^{-1}(V))= f_{\ast}(P)(V)\). This yields a natural transforamtion $f \Rightarrow g$ in $\geom(Sh(X),Sh(Y))$.
From the construction above we obtain a functor \(\mathbf{Top}(X,Y) \rightarrow \mathbf{Geom}(Sh(X),Sh(Y))\).
As a consequence, it is generally too much to expect diagrams in Top to commute on the nose”, i.e up to equality { commuting is usually only up to isomorphism. In broad terms, this is because in a category equality between ob jects is less important that isomorphism
$\x$
