Is the inclusion map always smooth?

This maybe a very silly question. But I am quite confused.

If $M$ is a smooth manifold, and $A\subset M$ is a submanifold, then is the inclusion map $i:A\longrightarrow M$ smooth?

My guess is that it might not be smooth. Because the manifold structures might not be compatible. But I am not able to precisely say this. I will be happy if someone helps!

As explained in the comments, smoothness is guaranteed by the definition of submanifold. The concern that "the manifold structures might not be compatible" is reasonable, but submanifolds are defined so that compatibility holds. Here is a non-example: let $M$ be the curve $y=|x|$ in the plane, considered as the image of $\mathbb R$ under $f(x) = (x,|x|)$. We can make $M$ a smooth manifold by declaring $f$ a diffeomorphism. Now $f:\mathbb R\to M$ is smooth but $f:\mathbb R\to\mathbb R^2$ is not; the smooth structures are not compatible. But of course, $M$ is not a submanifold of $\mathbb R^2$ here.

The definition you have (which is the definition of an immersed submanifold) has the smoothness of the inclusion map built in.