# Is this correlation between distance matrices?

I have a set of objects. I have calculated two distance matrices: $$X$$ defining distance between each objects pair using metric $$f1$$, and $$Y$$ -- using metric $$f2$$. Now, I would like to understand if two objects are similar according to metric $$f1$$, then they are also similar according to metric $$f2$$. How can I do it?

For instance, $$f1$$ could say whether two objects have similar color, and $$f2$$ --- whether two objects have similar size. But metrics can be anything. For instance, we could talk about articles, $$f1$$ could be Jaccard distance measuring how many tags both articles share, and $$f2$$ could be euclidian distance measuring distance between word vectors of two articles. Now I would like to understand if two objects of blueish objects tend to be big, or whether articles tagged with "racism" have similar content.

Am I asking about correlation? How can I calculate it between $$X$$ and $$Y$$?