Let $J,C$ be categories. Assume $J$ is small and has an initial object $j_0$, and $C$ is cocomplete.
Let $F',F \in \mathrm{Obj}(C^J)$ be diagrams/functors $J\rightarrow C$, and suppose $\eta : F' \rightarrow F$ is a natural transformation of diagrams. This induces a morphism $\operatorname{colim}\eta : c' \rightarrow c$, where $c' = \operatorname{colim}F'$, $c = \operatorname{colim}F$.
Let $a \in \mathrm{Obj}(C)$ and a morphism $f_0 : a \rightarrow F(j_0)$ and a morphism $\tilde{f} : a \rightarrow c'$, such that the composition $a\xrightarrow{f_0}F(j_0) \rightarrow c$ equals $a \xrightarrow{\tilde{f}} c' \xrightarrow{\operatorname{colim}\eta} c$.
I'm pretty sure in general, there does not exist $j' \in \mathrm{Obj}(J)$ and a morphism $a \xrightarrow{f'} F'(j')$ such that the composition $a\xrightarrow{f'}F'(j')\rightarrow c'$ equals $\tilde{f}$.
However, do such $j'$ and $f'$ as in the above line exist if $C$ is the category $\mathrm{Set}$?
Also I'm mainly considering the case where $J$ is the subcategory of $\mathrm{Open}(X)^{\mathrm{op}}$ consisting of open sets containing some base point $x \in X$, and $F$ is a $C$-valued sheaf on $X$. Does the above hold in this case? If $C = \mathrm{Set}$ also?
Below is a diagram of the scenario:
Note:
- I accidentally wrote uppercase "$A$" in the picture, but typed lowercase "$a$" in the question.
- The leftmost vertical arrow should be labeled $\operatorname{colim}\eta$, not $\eta$.
No. Take $F$ be the constant functor with value $c'$, and $\eta\colon F'\to F$ the natural transformation corresponding to the universal cocone. Then the conndition that $a\to F(j_0)\to c$ equals $a\to c'\to c$ holds vacuously for any morphism $a\to c'=c$.
The condition that for any morphism $a\to c'$ there exists $j'$ so that $a\to c'$ factors as $a\to F(j')\to c'$ asserts that the cocone $\mathrm{Hom}(a,F')\to\mathrm{Hom}(a,c')$ is colimiting, i.e. that the functor represented by $a$ preserves the colimit of $F$. This is a special condition. For example, the representable functor preserves all filtered colimits if and only if $a$ is a finite set.
It preserves all colimits of diagrams with an initial object if and only if it is a singleton or empty. To see this, take $c'=\{0,1,2\}$, $a\to c'$ surjective onto $\{1,2\}$, and write $c'$ as the pushout of $\{0\}\to\{0,1\}$ along $\{0\}\to\{0,2\}$. Neither of the composites $\{0\}\to\{0,1,2\}$, $\{0,1\}\to\{0,1,2\}$, and $\{0,2\}\to\{0,1,2\}$ are surjective onto $\{1,2\}$, so $a\to\{0,1,2\}$ does not factor through any of them.