The injective Čech model structure on simplicial sheaves on $C$ is the unique model structure on $sSh (C)$ with the following properties:
The weak equivalences are the morphisms in $sSh (C)$ that are weak equivalences in $[C^{op}, sSet]_{\check{C},inj}$.
The cofibrations are the monomorphisms.
The fibrations are the morphisms in $sSh (C)$ that are fibrations in $[C^{op}, sSet]_{\check{C},inj}$.
Definition
The projective Čech model structure on simplicial sheaves on $C$ is the unique model category structure on $sSh (C)$ with the following properties:
The weak equivalences are the morphisms in $sSh (C)$ that are weak equivalences in $[C^{op}, sSet]_{\check{C},proj}$.
The trivial fibrations are the morphisms in $sSh (C)$ that are trivial fibrations in the projective Čech model structure on $[C^{op}, sSet]_{\check{C},proj}$, i.e. the componentwise trivial Kan fibrations.
Constructions
To construct the injective Čech model structure on $sSh (C)$, we use the following facts:
The inclusion $sSh (C) \hookrightarrow [C^{op}, sSet]$ is fully faithful.
The left adjoint (i.e. sheafification) $[C^{op}, sSet] \to sSh (C)$ preserves monomorphisms, and the class of monomorphisms in $sSh (C)$ is closed under pushouts, transfinite composition, and retracts.
The adjunction unit is a natural weak equivalence with respect to the (injective) Čech model structure on $[C^{op}, sSet]$: see Theorem A.2 in DHI.
We may then apply Kan’s recognition principle for cofibrantly generated model structures to transfer the injective Čech model structure from $[C^{op}, sSet]$ to $sSh (C)$. By construction, the sheafification adjunction becomes a Quillen equivalence.