Whilst studying some category theory, I was blindly using the notation $\lim\_\leftarrow D$ for the limit of a diagram $D$ in some category $\mathcal{C}$ (as they are notated in MacLane, Awodey, my lectures etc.).

Why does the arrow go from right to left?

Is it because we think of the limit of a diagram as sitting on the left of the diagram (where the arrows in our diagram go from left to right)?

I sometimes think of a terminal object as being to the far right of a category, because everything can map into it. If limits had an arrow from left to right (like in analysis), I think I could justify this by saying "Well a limit is a terminal object in the category of cones, so it sits on the right". This makes me think that my 'reason' above is nonsense.

Any thoughts are appreciated. Thanks!