The projective span model computes the homotopy-pushout mapping property

Cofibrancy of the replaced span. In the projective model of [F1] the free span at its initial vertex is $(X\to X,X)$ and the two other free spans carry $X$ only at the respective vertex. Attaching a generating cell at the initial vertex pushes out its old initial object together with both legs, preserving the cofibrancy of both legs, while attaching at an endpoint composes that leg with a cofibration. Beginning with the initial span, passing through cell sequences and closing under retracts therefore shows that every projectively cofibrant span $A_0\to B_0$, $A_0\to C_0$ has $A_0$ cofibrant and both legs cofibrations. In particular $B_0,C_0$ are cofibrant and $D_0=B_0{\coprod }_{A_0}{C}_0$ is cofibrant as a pushout of cofibrations along a cofibrant base.

The strict pullback of mapping spaces. For fibrant $Y$ the enriched pushout identity gives $\mathrm{M}\mathrm{a}\mathrm{p}(D_0,Y)\cong \mathrm{M}\mathrm{a}\mathrm{p}(B_0,Y){\times }_{\mathrm{M}\mathrm{a}\mathrm{p}(A_0,Y)}\mathrm{M}\mathrm{a}\mathrm{p}(C_0,Y)\mathrm{.}$ Every displayed mapping space is Kan by [F1] and both maps to $\mathrm{M}\mathrm{a}\mathrm{p}(A_0,Y)$ are Kan fibrations by the corner axiom, so the strict pullback is a model for the homotopy pullback.

Comparison with the path homotopy pullback. Given a Kan fibration $f\text{\hspace{0.17em}⁣}:E\to T$ and any map $g\text{\hspace{0.17em}⁣}:F\to T$, put $P=E{\times }_{T}F$ and let $H$ be the set of triples $(e,z,\gamma )$ with $\gamma \text{\hspace{0.17em}⁣}:\mathrm{\Delta }[1]\to T$, $\gamma (0)=f(e)$ and $\gamma (1)=g(z)$; the constant-path map $i\text{\hspace{0.17em}⁣}:P\to H$ is a monomorphism. Over $H\times \mathrm{\Delta }[1]$ lift the path $\gamma$ through $f$ with prescribed initial value $e$, requiring the lift to be constant on $i(P)\times \mathrm{\Delta }[1]$; the inclusion $(H\times \{0\})\cup (i(P)\times \mathrm{\Delta }[1])\subset H\times \mathrm{\Delta }[1]$ is the pushout product of $i(P)\subset H$ with the horn $\{0\}\subset \mathrm{\Delta }[1]$, hence anodyne by [F2], so the simultaneous lift exists. Let $h$ be that lift and set $r(e,z,\gamma )=(h(1),z)\in P$; the prescribed constant lift gives $ri=\mathrm{i}\mathrm{d}$. Homotoping ${\mathrm{i}\mathrm{d}}_H$ to $ir$ is given at time $t$ by the first coordinate $h(t)$, the unchanged second coordinate $z$, and the path $s\mapsto \gamma (\max (s,t))$, which is a simplicial map because max is order-preserving on $[1]\times [1]$; at $t=0$ the original triple is recovered, at $t=1$ it is $ir$, and the homotopy is constant on $i(P)$. Hence $i$ is a simplicial deformation equivalence and the strict pullback of step 2.1 agrees with the homotopy pullback.

Derived invariance and scope. The derived enriched colimit adjunction of [F1] together with the replacement invariance of [F3] makes the mapping object $\mathrm{M}\mathrm{a}\mathrm{p}(D_0,Y)$ independent of the chosen projective replacement in the model homotopy theory, so it computes the derived enriched colimit mapping property. The construction does not by itself identify, for an arbitrary noncofibrant base, the strict model category of algebras with the homotopy-coherent undercategory, and no such unrestricted base-change or localization theorem is claimed; for a cofibrant base and cofibrant span the explicit comparison above is available.