Reading the article on Quillen bifunctors on the Nlab (https://ncatlab.org/nlab/show/Quillen+bifunctor) I stumbled upon the following claim: Let $\mathcal{C}$ be a combinatorial simplicial model category and let $\mathcal{D}$ be a small category. Furthermore, let $F \colon \mathcal{D}\to \mathcal{C}$ be a functor and $\star \colon \mathcal{C}^\text{op} \to \text{sSet}$ be the constant functor on the terminal object. Denote by $\text{sSet}^{\mathcal{D}^\text{op}}_\text{proj}$ and $\mathcal{C}^{\mathcal{D}}_\text{inj}$ the obvious categories endowed with the projective and injective model structures, respectively. The claim made is then that the homotopy colimit of $F$ may be given by a coend formula: $$\mathop{\text{hocolim}}\limits_{\mathcal{D}} F = \int\limits^\mathcal{D} Q_\text{proj}(\star) \cdot Q_\text{inj}(F)$$ where $Q_\text{proj}$ and $Q_\text{inj}$ are the respective cofibrant replacement functors. Apparently this formula follows from the fact that $$\int\limits^\mathcal{D}(-\cdot -): \text{sSet}^{\mathcal{D}^\text{op}}_\text{proj} \times \mathcal{C}^{\mathcal{D}}_\text{inj}\to \mathcal{C}$$ where $\cdot \colon \text{sSet} \times \mathcal{C} \to \mathcal{C}$ denotes the copower, is a left Quillen bifunctor.
How exactly does one deduce the homotopy colimit formula from the fact that the above is a Quillen bifunctor, assuming that the homotopy colimit is defined to be the left derived functor of $\text{colim}$?
Following the hints in the comments, I have come up with the following: Denote by $\odot \colon \text{sSet} \times \mathcal{C}\to \mathcal{C}$ the tensoring of $\mathcal{C}$ over $\text{sSet}$. We then have $$\mathcal{C}(\Delta^0 \odot X, Y) \cong \text{Map}(\Delta^0,\mathcal{C}(X,Y))\cong \mathcal{C}(X,Y)$$ for all $X,Y \in \mathcal{C}$ (where $\text{Map}(-,-)$ is the internal hom in $\text{sSet}$) and hence $\Delta^0 \odot - \cong 1_\mathcal{C}$. This then implies $\int\limits^\mathcal{D} \Delta^0 \odot (-) \cong \mathop{\text{colim}}\limits_\mathcal{D}$. This is a left Quillen functor, so we may derive it to find: $$\mathop{\text{hocolim}}\limits_\mathcal{D}=\mathbb{L}\int\limits^\mathcal{D} \Delta^0 \odot (-) =\int\limits^\mathcal{D} \Delta^0 \odot Q_\text{inj}(-)$$ But then since $$\int\limits^\mathcal{D} (-) \odot (-)$$ is a left Quillen bifunctor, the weak equivalence $Q_\text{proj}(\Delta^0) \overset{\simeq}{\to} \Delta^0$ induces a natural weak equivalence $$\int\limits^\mathcal{D} Q_\text{proj}(\Delta^0)\odot Q_\text{inj}(-) \simeq \int\limits^\mathcal{D} \Delta^0 \odot Q_\text{inj}(-)$$ Hence indeed, $$\mathop{\text{hocolim}}\limits_\mathcal{D} \simeq \int\limits^\mathcal{D} Q_\text{proj}(\Delta^0)\odot Q_\text{inj}(-)$$