What they need to know is:

- what the risks are

- what they have to do about the risks

Whether or not reading the COSHH assessment will convey this will depend on how the COSHH assessment is documented.  There may be a better way to convey it.

I'm not sure what you mean by having a COSHH assessment as well as a risk assessment for the activity.   What do the COSHH assessment and the risk assessment cover that is different?  A COSHH assessment is in theory a type of risk assessment, one for an activity that involves the use of hazardous substances.

So what is the content of your COSHH assessments apart from stating the hazards of the substances?  Do they just replicate parts of the safety data sheet (this is all too often the case) or do they contain information about the task and about how exposure to the hazardous substances is prevented or limited during the task?  And is that information missing from the task risk assessment?

I disagree that COSHH assessments are likely to be useful to a first aider.  What the first aider usually needs is the safety data sheet for the substance someone has been exposed to.  This not only has a first aid section in it, but other medical information that although the first aider will not use it, they will want to pass on if medical treatment is required.  A paramedic or doctor doesn't want to see a company's COSHH assessment - they want to see a safety data sheet, which is the authoritative document about the substance.

But to go back to your original question. It sounds as if your risk assessments contain instructions for those doing the task.  If there are any instructions relating to the use of hazardous substances, it would make sense to convey these at the same time using the same document as conveying the other safety instructions.

Kate is of course correct: all risk assessments are task based including COSHH ones. So in your task risk assessment will need to mention that the Loctite glue that is being  used to  fix your bolts is hazardous: mainly you don’t want to get it in your eye. The risk of that happening depends on how it is being applied. That is one risk assessment.  First aiders need information, on how to treat various injuries. That information needs to be to the point and give them exactly what they need to know. They don’t need to see the SDS which has lots of superfluous (to the first aider) information- boiling point, vapour pressure, etc? For Loctite in the eye the question is do you try to remove if from the eye as soon as possible or do you get them to A&E as soon as for them to try to remove it?  

To AK's point about first aid, section 4 of a Loctite safety data sheet (I've looked at many of these) contains detailed information about what to do if someone gets it in their eye, or indeed on their skin or in their mouth.  That, and section 2 with the hazards, are the only bits of the safety data sheet that the first aider will make use of.  The reason I advocate giving them the whole thing is so that it can be passed on to medical professionals.