When deriving the lens maker's equation I faced the problem of understanding the logic behind this reasoning "image formed by the first surface of a lens serves as the object for the second surface."
It is important to understand what an image is and where it is positioned.
Remember that your definition has to work for things you see directly as well as those that are viewed through optics. It should also work for both real and virtual images.
The version I use in class is
An image is a collection of rays originating from a single object that appear to come from a common location in space (that may or may not be the actual object), and the image is said to be positioned at their apparent origin.
When the light leaves the original object the rays are dispersing from the object. That is the "object" used by the first surface which bends the light and generates an image.
"Generates an image" means that after the first surface the rays now appear to come from a different place.
It is those rays that are incident on the second surface so as far as the second surface is concerned the object that it images is the new set of rays with their new "position".
The only surprise here is that this approach works even when the image of the first surface is downstream from the second surface, but if you trace the rays you'll see that it all works out.