Given a function $f : X \rightarrow Y$ and a set $A \subseteq X$, there's at least two possible ways of interpreting the direct image $f(A),$ as explained here. In the notation of that question, write $f_*(A)$ for the usual direct image, and $f_\diamond(A)$ for the variant. The definitions are as follows.
$$f_*(A) = \{b \in Y \mid \exists a \in X : a \in A \,\wedge\, f(a)=b\}$$ $$f_\diamond(A) = \{b \in Y \mid \forall a \in X : a \in A \,\vee\, f(a) \neq b\}$$
Note that in the case of injections bijections, we have $f_* = f_\diamond$. Observe also that these definitions are dual, in the sense that $[f_\diamond(A^c)]^c = f_*(A)$ and $[f_*(A^c)]^c = f_\diamond(A).$
Now in lattice theory, we have the following definition: given complete lattices $L$ and $M$ and a function $f : L \rightarrow M$, we say that $f$ is a complete lattice homomorphism iff $$f\left(\bigwedge A\right) = \bigwedge f_*(A), \quad f\left(\bigvee A\right) = \bigvee f_*(A)$$
Why is it defined this way? What would happen if we instead required:
$$f\left(\bigwedge A\right) = \bigwedge f_\diamond(A), \quad f\left(\bigvee A\right) = \bigvee f_\diamond(A)$$
??? Note that, whenever $f$ is injective bijective, these two definitions coincide.