# What is the smoothness of a family of diffeomorphisms $t\mapsto \psi_t \in \text{Diff}(M)$ ? And how to interpret it intuitively?

First of all, we have to give the group $\text{Diff}(M)$ of all diffeomorphisms on $M$ a smooth-manifold structure. (To see this, it may be helpful to consider a easier problem: how to give the group $\text{Homeo}(M)$ of all homeomorphisms on $M$ a topology structure. I guess it is possibly related to compact-open topology on the sets of continuous maps between two topological spaces.)

In addition, the local flow $\phi_t$ of a smooth vector field $X\in \mathcal X(M)$ may be a good example of a smooth family of diffeomorphisms. With this in mind, I was wondering if the following statement is a equivalent condition of the smoothness of a family $\phi_t\in \text{Diff}(M)$: for every $p \in M$ the path $t \mapsto \phi_t(p)$ is smooth in $M$.

I am reading Mcduff & Salamon's book on symplectic topology. The following text and proposition is the place where I encounter my problem (Page 83).

I would expect that for the purpose you are aiming at it is sufficient that $\psi$ is smooth as a map $[0,1]\times M \rightarrow M$ (where the first factor has a smooth structure in a natural way) with the additional constraint that, for each $t$, $\psi_t$ is a diffeomorphism (or symplectomorphism). This is, however, not the structure of a differentiable manifold on $\text{Diff}(M)$, and for most cases this is not needed.