Let $I$ be the category with objects points of $[0,1]$ with unique morphism for every pair of objects. Let $C$ be a complete category, suppose that there is an isomorphism $m: a \to b$, $m \in C$ when does there exist a functor $F: I \to C$, with $I(0)=a$, $I(1)=b$, and such that the following holds. Let $D(t)$ denote the directed system $$t_1 \mapsto t _{2} \ldots \mapsto t_i \mapsto \ldots $$ for $t_i \in I$, for all $i$, and $t_i$ converges to $t$. Then $colim_{D(t)} F = F(t)$, for every $t$ and every $D(t)$ as above.

Edit: The original version of the question was trivial as answered below. I added the "continuity" assumption in the form of the colimit condition.

Edit2: Simplified the colimit condition and changed $I$.

any$b$ with a map $m:F(\tau)\to b$ and define $F(t)=b$, $F(\tau\leq t)=m$ for $t>\tau$, with $F(t\leq t')=\mathrm{id}_B$. Your definition imposes an analogue of lower semicontinuity, so maybe you also want to add the dual criterion that $F$ commutes with limits from the right. $\endgroup$