Let $A$ be a lattice, then $A$ is a frame iff it is a complete Heyting algebra. See Topology via logic

Also check Theorem 3.5.2, that a frame is a complete lattice and hence has infinite meets.

A map $f : A \to B$ is a frame homomorphism iff

• preserving finite meets $f(a \wedge b) = f(a) \wedge f(b)$
• preserving arbitrary joins $f(\bigvee_{i\in I} a_i) = \bigvee_{i\in I} f(a_i)$

A map $f : A \to B$ is a cHa homomorphism iff

• preserving arbitrary meets $f(\bigwedge_{i\in I} a_i) = \bigwedge_{i\in I} f(a_i)$
• preserving arbitrary joins $f(\bigvee_{i\in I} a_i) = \bigvee_{i\in I} f(a_i)$
• monotone (preserving Heyting implication) $f(a \Rightarrow b) = f(a) \Rightarrow f(b)$

cHa homomorphism has a left adjoint and a right adjoint