one commutes up to the specified homotopy $h{\colon}[0,1]\times X\to B$ relative to $A$. We will not pursue this approach any further (in particular, we will not define the right notion of homotopy of such pairs) and choose an equivalent, technically less demanding alternative, which consists in replacing the map $\varphi$ by a homotopy equivalent fibration.

To this end, we factor $\varphi{\colon}Y\to B$ as a weak homotopy equivalence $j{\colon}Y{\xrightarrow{\sim}} Y'$ followed by a fibration $\varphi'{\colon}Y'{\mathbin{\:\!\!\xymatrix@1@C=15pt{{}\ar@{->>}[r]^{} & {}}}}B$ (in the simplicial setup, see Lemma \[l:fibrant\_replacement\]). We define a *solution* of the considered generalized lifting-extension problem to be a solution $\ell{\colon}X\to Y'$ of the lifting-extension problem $${\xy *!C\xybox{\xymatrix{
A \ar[r]^-{f} \ar@{ >->}[d] & Y \ar[r]^-j & Y' \ar@{->>}[d]^-{\varphi'} \\
X  \ar[rr]_-{g} \ar@{-->}[rru]^-\ell & & B
}}\endxy}$$ If $\varphi$ was a fibration to begin with, we naturally take $Y=Y'$ and $j={\mathop\mathrm{id}\nolimits}$, and then the two notions of a solution coincide. With some abuse of notation, we write $[X,Y]_B^A$ for the set $[X,Y']_B^A$ of all homotopy classes of solutions of the above lifting-extension problem.

We remark that $Y'$ is used merely as a theoretical tool – for actual computations, we use a different approximation of $Y$, namely a suitable finite stage of a Moore–Postnikov tower 