Following Dan Schmidt's post 5, it seems that if \$$f_{\ast}\$$ has a left adjoint \$$g_{\ast}\$$, then \$$f\$$ must be injective.
Otherwise, if \$$f\$$ is not injective, there are elements \$$a_1, a_2 \in A, a_1 \neq a_2\$$ such that \$$f(a_1) = b = f(a_2)\$$.
Then, \$$f_{\ast}\$$ sends both \$$\\{a_1\\}\$$ and \$$\\{a_2\\}\$$ to \$$\\{b\\}\$$.
The left adjoint \$$g_{\ast}: PY \to PX\$$ must then send \$$\\{b\\}\$$ to something at most \$$\\{a_1\\} \wedge \\{a_2\\} = \emptyset\$$, which must be \$$\emptyset\$$ itself.
But then we would have \$$\emptyset = g_{\ast}(\\{b\\}) \leq \emptyset\$$ while \$$\\{b\\} \nleq f_{\ast}(\emptyset) = \emptyset\$$, contradicting our hypothesis.