You might hear a PT say, "don't drink coffee after training because it will stimulate the release of stress hormones and stop you from recovering from training, which is bad because you want rest and digest mode". 

While there may be an element of truth to this. There are a few things wrong with it. 

The level of difference is so small it's not worth even mentioning in almost all situations. If you train first thing and like a coffee after, go for it! You have bigger things to worry about. 

We know consuming caffeine pre-training is a good idea to drive better training outcomes leading to 'better' adaptations. However, caffeine takes several hours to clear the blood levels after ingestion. So even if you didn't drink a coffee after your morning workout, the one you drank before it is still lingering for after you finish well into the 'post workout window'. 

Post-training caffeine might be a good strategy for athletes. Caffeine has been shown to promote greater rates of glycogen resynthesis after exercise when ingested with carbohydrate. Basically, this means if you train hard, then drink coffee and consume carbohydrates together you'll store more glucose in your muscle and at a faster rate. While this isn't massive for training adaptations and certainly not important for most folks. It is worth noting for best recovery, especially if you're doing repeated bouts of hard competition level activity within very close time proximity of each other. [1,2]

Take home points. 
Drinking coffee, and consuming caffeine after training isn't something to be afraid of. It may even be a great idea for certain situations.  

[1] High rates of muscle glycogen resynthesis after exhaustive exercise when carbohydrate is coingested with caffeine.

[2] Caffeine ingestion does not impede the resynthesis of proglycogen and macroglycogen after prolonged exercise and carbohydrate supplementation in humans.