Proposition 11.10.7.13. Let $f: X \hookrightarrow Y$ be a monomorphism of simplicial sets. The following conditions are equivalent:
The morphism $f$ is left anodyne.
The morphism $f$ is a covariant equivalence relative to $Y$.
For every morphism of simplicial sets $Y \rightarrow S$, the morphism $f$ is a covariant equivalence relative to $S$.
Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 11.10.6.6, and the implication $(2) \Rightarrow (3)$ from Remark 11.10.7.7. We will show that $(3)$ implies $(1)$. For this, we must show that every lifting problem
admits a solution, provided that $q$ is a left fibration of simplicial sets (Corollary 4.2.4.10). To prove this, it will suffice to show that the natural map $\theta : \operatorname{Fun}_{/S}(Y,Z) \rightarrow \operatorname{Fun}_{/S}(X,Z)$ is surjective on vertices. In fact, our assumption that $f$ is a monomorphism and a covariant equivalence relative to $S$ guarantees that $\theta $ is a trivial Kan fibration of simplicial sets (Corollary 11.10.7.12). $\square$