Q:

Remembering arrows' directions in basic Category Theory

Is there an easy way of remembering the direction of arrows between morphisms in Categories?  
The direction of arrows so confuses me: products and co-products, (EDIT- Also, pull-backs, pushouts, contra/co-variant functors) and their universal mapping properties.  I have to look back into Lang's Algebra and revise it every time.  
Please tell if the question is not appropriate. 

A:

I think that the best way to remember these things is to have a good example of each of these constructions in your head.  If you remember the direction of the arrows for them, and if the example is "natural" enough the direction will be obvious, you will have the direction in the general case.  Just watch out, sometimes two constructions will yield the same things in specific examples so you want to have examples that distinguish between  the various notions.  An example of this is that finite products and finite co-products have a habit of being the same in categories in which I search for examples (additive categories like categories of representations). 

A:

Aside from having a good clear example in mind: Cartesian products for products, disjoint unions for coproducts, pullback bundles for pullbacks,... one way I've found useful to remember directions of arrows involving universal objects is that arrows should factor through the universal object. 

A:

For most universal properties the universal object is the smallest one which satisfies a certain property.  For instance the product is the "smallest object" which allows you to complete its diagram, so anything which also satisfies it will project onto the product (this requires remembering the maps for the product are projections and not injections).  Likewise the coproduct is supposed to be the "smallest object" allowing you to complete its diagram, so there should be an injection from the coproduct to anything else.  The intuition for them being small is because the maps in the universal property are unique.