One of the definitions of the image of a morphism $f : c \to d$ is in terms of universal subobjects – i.e. universal monomorphisms – through which $f$ factors.

A morphism $f : c \to d$ in $C$ is called a homotopy monomorphism if the universal morphism $Id \times Id : c \to c \times^h_d c$ into its homotopy pullback along itself is an isomorphism in the homotopy category.

The homotopy image of $f$ is a factorization of $f$ into a cofibration $c \to f(c)$ followed by a homotopy monomorphism $f(c) \to d$

such that for any other such factorization $c \to e \to d$ there exists a unique morphism $f(c) \to e$ in the homotopy category making the obvious triangles commute.