Are there cases where the two notions coincide?
By sheaf I mean a pre-sheaf ($\mathcal{C}^{op} \rightarrow \mathcal{Set}$) satisfying the sheaf condition. The sheaf condition says that, for every open cover of $U \subseteq X$ = opens sets of $\mathcal{C}$, there exists a unique glued section $s$ over $U$. A glued section $s$ is such that every pair of components $s_i, s_j$ agree on their intersection $U_i \cap U_j$ on the base.
By a fibration $\pi: \mathcal{E} \rightarrow \mathcal{B}$ I mean for every object C of $\mathcal{E}$ and morphism $\gamma : I \rightarrow \pi C$ in $\mathcal{B}$, there is a Cartesian morphism $c: E \rightarrow C$ above $\gamma$. We call $c$ the Cartesian lifting of $\gamma$. A Cartesian morphism is such that for each map $c$ in $\mathcal{E}$, one can form a "triangle" of maps in $\mathcal{E}$ and a corresponding triangle of maps in $\mathcal{B}$ and they commute and agree with the projection $\pi$.
The lifting property says that if $\pi: \mathcal{E} \rightarrow \mathcal{B}$ is continuous and if $X$ is a "nice" topological space (ie, locally path-connected) then a homotopy $H : X \times I \rightarrow \mathcal{B}$ can be lifted to a homotopy $\tilde{H} : X \times I \rightarrow \mathcal{E}$.
My question is: Is the lifting property (essentially) the same as the sheaf condition? Or are they completely different?
