The group of homotopy self-equivalences is a Lax functor

The group $\text{E}(X)$ of homotopy self-equivalences of a topological space $X$ is a well-known group in homotopy theory and has been studied by many people since it was first introduced in the late 1950s. $\text{E}$ is not a functor in the usual sense. In this paper we show that $\text{E}$ is a Lax functor from the category $\mathcal{T}op$ of topological spaces to a strict $2$-category $\text{op}{Corr}_{\mathcal{G}r}$ of \emph{correspondences} of groups.