Let $(X_t)$ a stochastic process and $(\mathcal F_t)$ its natural filtration. For me, if for all Borel set $A$, $$\mathbb P\{X_t\in A\mid \mathcal F_s\}=f(t,s,A,X_s),\tag{1}$$ then $$\mathbb P\{X_t\in A\mid \mathcal F_s\}=\mathbb P\{X_t\in A\mid X_s\}.$$
Therefore, $(X_t)$ is a Markov process.
Q1) Am I right ?
But in my lecture, they want $$\mathbb P\{X_t\in A\mid \mathcal F_s\}=P_{t,s}(A,X_s),$$ where $(P_{t,s})$ is a familly of transition function.
Q2) So, I guess that $(1)$ is not enough to be a markov process, is it ? But maybe that the function $f(t,s,\cdot ,\cdot )$ is automatically a transition function whenever $(1)$ hold ? If not, could someone provide a counter example ?