Definition 1.5.4.3. Let $\operatorname{\mathcal{C}}$ be a category and suppose we are given a morphism $f: A \rightarrow B$ and $g: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$. We will say that *$f$ is weakly left orthogonal to $g$* if, for every pair of morphisms $u: A \rightarrow X$ and $v: B \rightarrow Y$ satisfying $g \circ u= v \circ f$, the associated lifting problem

admits a solution (that is, there exists a map $h: B \rightarrow X$ satisfying $g \circ h = v$ and $h \circ f = u$). In this case, we will also say that *$g$ is weakly right orthogonal to $f$*.

If $S$ and $T$ are collections of morphisms of $\operatorname{\mathcal{C}}$, we say that *$S$ is weakly left orthogonal to $T$* if every morphism $f \in S$ is weakly left orthogonal to every morphism $g \in T$. In this case, we also say that *$T$ is weakly right orthogonal to $S$*. In the special case where $S = \{ f\} $ is a singleton, we abbreviate this condition by saying that *$f$ is weakly left orthogonal to $T$*, or *$T$ is weakly right orthogonal to $f$*. In the special case $T = \{ g\} $ is a singleton, we abbreviate this condition by saying that *$g$ is weakly right orthogonal to $S$*, or *$S$ is weakly left orthogonal to $g$*.