Let $C$ and $I$ be categories and $I$ is filtered. Let $F$ be an inductive system indexed by $I$ in $C$. Then we have the ind-object
$$X\to \varinjlim\limits_{i\in I} \mathrm{Hom}_C(X, F(i)),$$
which Kashiwara and Schapira denote by $``\varinjlim\limits_{I}`` F$. According to Proposition 1.11.6 of their "Sheaves on Manifolds", an object $L\in C$ represens $``\varinjlim\limits_{I}`` F$ if and only if $\forall i\in I$ there are morphisms $\rho_i: F(i)\to L$ and there is an object $i_0\in C$ and a morphism $f:L\to F(i_0)$ such that:
- $\forall s:i\to j$ we have $\rho_j \circ F(s)=\rho_i$,
- $\rho_{i_0}\circ f=\mathrm{id}_L$,
- $\forall i\in I$ $\exists j\in I$ and morphisms $s:i\to j$ and $t:i_0\to j$ such that $F(t)\circ f\circ \rho_i= F(s)$.
I can't wrap my head around what this criterion actually means. How should one think of these ind-objects that actually exist in $C$? It follows that if $``\varinjlim\limits_{I}`` F$ is represented by $L$, then $L$ also represents $\varinjlim\limits_{I} F$. Right now I think that the criterion says that the colimit is also the colimit as an ind-object if it is also "determined by a bounded part of $I$". Is this correct, or could you suggest a better way to think of this?