$cl$ is a closure operator. It is monotone because enlarging $\mathcal{F}$ can only shrink $≡[\mathcal{F}]$, and extensive because $≡[\mathcal{F}]$ determines the homomorphism counts from every member of $\mathcal{F}$ by definition. It is idempotent because $≡[\mathcal{F}]$ and $≡[cl \mathcal{F}]$ are the same relation.