Inner automorphisms Inn(D4). We need to show that elements of $Inn(D_4)$ are distinct , where , $Inn(D_4)= \phi_{{R_0}} , \phi_{{R_{90}}} , \phi_{H} , \phi_{D}$. Is it sufficient to construct a Cayley table for the elements of $Inn(D_4)$ , and thus conclude that any composition of these elements belongs to $Inn(D_4)$ only , and thus the elements are distinct ? Will that be OK ?

The question is not whether the set is closed under multiplication. The question is to show that they are all different.

The automorphisms are functions. Functions with the same domain and range are the same if they have the same values. To show two of the functions aren't the same, find an element on which they yield different values. Do this every pair.