WITH in this situation is probably best described as a preposition ... and here it gets tricky for me... "indicating that something is followed by something else!" !
With the warmer days now followed by an evening chill, I feel autumn coming.
With the success of my presentation followed by my failure in the exam, I feel I'm on life's rollercoaster once again.
I think it may be useful to think of it as an expression. It indicates a connection, in this case of one thing causing or following another.
With all this confusion over WITH, followed by my even more confusing explanation, we're pretty mixed up, aren't we!