In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line 3-line 6:
for a map $p:E\to B$ and $b\in B$, there is a natural map $p^{-1}(b)\to F(p,b)$ where $F(p,b)$ is the homotopy fibre.
I find the definition of homotopy fibre on Hatcher's Algebraic Topology, page 407-409:
Let $p: E\to B$ be a map. Let the space $$ E_p=\{(a,\gamma)\in E\times \text{Map}([0,1],B) \mid \gamma(0)=p(a)\} $$ be topologized as a subspace of $E\times \text{Map}([0,1],B)$. For every $b\in B$, the homotopy fibre of $p$ at $b$ is defined to be the space $$ F(p,b)=\{(a,\gamma)\in E\times \text{Map}([0,1],B)\mid \gamma(0)=p(a), \gamma(1)=b\}. $$
Question:
How to define the natural map $$ p^{-1}(b)\to F(p,b)? $$ I want to give the natural map by sending $a\in p^{-1}(b)$ to $(a,\gamma_a)\in F(p,b)$. However, the path $\gamma_a$ from $p(a)$ to $b$ is not uniquely determined since $B$ may not be simply connected.