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 $I$.



Existence of "Continuous paths" in categories as directed systems
This is a bit of a boring answer, but such a functor always exists. We can use the functor that sends $0$ to $a$, every thing else in $[0,1]$ to $b$ and then sends the morphisms to $\mathrm{id}\_a$, $\mathrm{id}\_b$ or $m$ as appropriate.


