Consider a mapof (pointed) CW complexes $f:F\to E$, and a fibration $p:E\to B$. What data is necessary to say if $f$ is equivalent to the inclusion of the fiber $f':F'\to E$?
Clearly $F$ should be homotopic to $F'$, and the composition $p\circ f$ is nullhomotopic. For example, one can map $\mathbb Z$ to $\mathbb{R}$ by sending everything to $0$, then map $\mathbb R$ to the circle $\mathbb S^1$ according to the usual exponential. This is not the usual fiber (the copy of the integers in $\mathbb R$) but there is a homotopy between them (send $n$ from zero to $n$ at $n$ times unit speed).
Are these necessary conditions sufficient?